Non-stationary Research Articles

Recursive generalized gamma kernel density estimation for nonnegative dependent data

In this article, we consider the recursive generalized gamma (GG) kernel density estimator (KDE) for nonnegative dependent data from stationary α-mixing process. Asymptotic bias, variance and mean integrated squared error are provided. Simulation experiments from an autoregressive conditional duration model and a stochastic volatility model are conducted to compare the recursive GG KDE with non-recursive GG KDE. Two applications are provided for nonnegative time series data.

Thermal protection properties of profiled timber with peat-vermiculite insulation

Today, the use of environmentally friendly materials is becoming increasingly important. The construction of energy-saving buildings with enclosing structures made of insulated profiled timber, is one of the most promising areas in low-rise construction.Materials having minimum negative impact on the environment during their operation, include new construction insulation materials based on peat-vermiculite granules. The paper studies the main physical properties of wooden exterior walls made of profiled timber, in which peat vermiculite is used as insulation, thermal conductivity of which ranges from 0.05 to 0.06 W/(m·K). For this purpose, numerical simulation is performed for the heat-mass transfer in wooden enclosing structures made of insulated timber with three different connector options.The temperature and heat flow distributions are obtained as a result of modeling stationary thermal conductivity processes in timber structures. It is shown that plywood connectors do not affect the thickness temperature distribution in wooden walls with peat-vermiculite insulation. An increase in thermal conductivity of insulation has no significant effect on the temperature difference along the connector axis or outside the region of its temperature influence.It is found that the thickness of the peat-vermiculite insulation layer should be at least 0.176 m to ensure the compliance with the main requirements for thermal protection properties of enclosing structures in the climatic conditions of Tomsk. The greatest heat loss with a minimum insulation thickness is observed for the wall made of profiled insulated timber with T-shaped connectors. A high-strength timber with the additional central lamella has the best thermal protection with the lowest insulation thickness.

Research of stochastic stability of constructions parametric vibrations by the Monte-Carlo method

The stochastic parametric vibrations of the elastic systems behave to the statistical dynamics section of the nonlinear systems. Under stochastic parametric influence the vibrations of the elastic systems, which is a random process more frequent are not stabilized, but their amplitudes are fading or unlimited are growing. Therefore important is stability of stochastic vibrations, which can be examined as stability by probability, on average or by the moment functions of different orders. The mathematical models which describe the stochastic parametric vibrations of the elastic systems are building by analytical or numeral approaches. Researches of stochastic stability of parametric vibrations are executing by probabilistic methods. Stability of parametric vibrations of modern constructions under the action of operating random loads are not enough investigated. In the article the numeral method of research of dynamic stability of the elastic systems under stochastic parametric influence was presented. The mathematical model of stochastic parametric vibrations of construction in the form of a reduсed finite element model was formed. Matrixes of the reduced model were obtained using calculated procedures of finite element analysis software NASTRAN. The dynamic loading as stationary ergodic random process with the spectral density was presented in the form of a finite amount of harmonic functions. Stochastic stability of the constructions was investigated using the Monte Carlo method and the Runge Kutta method of direct numeral integration of the equations of reduсed model. Stochastic stability as stability by probability of appearance of tendency to fading or unlimited growth of amplitude of parametric vibrations was examined during the time interval in space of the random parameters of loading. Dynamic stability of a I-beam with corrugated wall under narrow-band parametric influence was investigated using the numeral method.

Characterising X-ray variability in light curves with complex sampling patterns: Application to the eROSITA south ecliptic pole survey

Aims : During its all-sky survey phase, the extended ROentgen Survey with an Imaging Telescope Array (eROSITA) X-ray telescope on board the Spectrum-Roentgen-Gamma (SRG) spacecraft scans through the ecliptic poles every 4 hours. This extensive data set of long-duration, frequent, and consistent observations of thousands of X-ray sources is ideal for a detailed long-term X-ray-variability analysis. However, individual observations are short, are separated by long but consistent gaps, and have varying exposure times. Therefore, the identification of variable sources and the characterisation and quantification of their variability requires a unique methodology. We aim to develop and evaluate variability analysis methods for eROSITA observations, focusing on sources close to the survey poles. We also aim to detect intrinsically variable sources at any count rate and quantify the variability of low-count-rate sources. Methods : We simulate eROSITA-like light curves to evaluate and quantify the effect of survey mode observations on the measured periodogram and normalised excess variance. We introduce a new method for estimating the normalised intrinsic variance of a source based on the Bayesian excess variance (bexvar) method. Results : We determine thresholds for identifying likely variable sources while minimising the false-positive rate, as a function of the number of bins, and the average count rate in the light curve. The bexvar normalised intrinsic variance estimate is significantly more accurate than the normalised excess variance method in the Poisson regime. At high count rates, the two methods are comparable. We quantify the scatter in the intrinsic variance of a stationary pink-noise process, and investigate how to reduce it. Finally, we determine a description of the excess noise in a periodogram caused by varying exposure times throughout a light curve. Although most of these methods were developed specifically for analysing variable active galactic nuclei in the eROSITA all-sky survey, they can also be used for the variability analysis of other datasets from other telescopes, with slight modifications.

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

In this paper, stationary random processes with fuzzy states are studied. The properties of their numerical characteristics—fuzzy expectations, expectations, and covariance functions—are established. The spectral representation of the covariance function, the generalized Wiener–Khinchin theorem, is proved. The main attention is paid to the problem of transforming a stationary fuzzy random process (signal) by a linear dynamic system. Explicitform relationships are obtained for the fuzzy expectations (and expectations) of input and output stationary fuzzy random processes. An algorithm is developed and justified to calculate the covariance function of a stationary fuzzy random process at the output of a linear dynamic system from the covariance function of a stationary input fuzzy random process. The results rest on the properties of fuzzy random variables and numerical random processes. Triangular fuzzy random processes are considered as examples.

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

Вданной работе изучены стационарные случайные процессы с нечеткими состояниями. Установлены свойства их числовых характеристик – нечетких ожиданий, ожиданий и ковариационных функций. Обосновано спектральное представление ковариационной функции – обобщенная теорема Винера–Хинчина. Основное внимание уделено задаче о преобразовании стационарного нечетко случайного процесса (сигнала) линейной динамической системой. Получены формулы, связывающие нечеткие ожидания (и ожидания) входных и выходных стационарных нечетко случайных процессов. Разработан и обоснован алгоритм вычисления ковариационной функции стационарного нечетко случайного процесса на выходе линейной динамической системы по ковариационной функции стационарного входного нечетко случайного процесса. Полученные результаты опираются на свойства нечетко случайных величин и числовых случайных процессов. Вкачестве примеров рассмотрены треугольные нечетко случайные процессы.

Estimation of stationary optimal transport plans

Abstract We study optimal transport for stationary stochastic processes taking values in finite spaces. In order to reflect the stationarity of the underlying processes, we restrict attention to stationary couplings, also known as joinings. The resulting optimal joining problem captures differences in the long-run average behavior of the processes of interest. We introduce estimators of both optimal joinings and the optimal joining cost, and establish consistency of the estimators under mild conditions. Furthermore, under stronger mixing assumptions we establish finite-sample error rates for the estimated optimal joining cost that extend the best known results in the iid case. We also extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem. Finally, we validate our convergence results empirically as well as demonstrate the computational advantage of the entropic problem in a simulation experiment.

On the existence of stationary threshold bilinear processes

This article investigates some statistical and probabilistic properties of general threshold bilinear processes. Sufficient conditions for the existence of a causal, strictly and weak stationary solution for the equation defining a self-exciting threshold superdiagonal bilinear SETBL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( SETBL\\right) $$\\end{document} process are derived. Then it is shown that under well-specified hypotheses the higher-order moments of the SETBL process are finite. As a result, the skewness and kurtosis indexes are explicitly computed. The exact autocorrelation function is derived with an arbitrarily fixed number of regimes. Also, the covariance functions of the process and its powers are evaluated and the second (respectively, higher)-order structure is shown to be similar to that of a linear process. This implies that the considered process admits an ARMA representation. Finally, necessary and sufficient conditions for the invertibility and geometric ergodicity of a SETBL model are established. Some examples illustrate the obtained theoretical results.

Spectral density estimation for random processes with stationary increments

AbstractSpectral density analysis plays an important role in studying a stationary random process on a real line. In this paper, we extend this discussion for the random process with stationary increments. We investigate the properties of the method of moments structure function estimation, and propose a nonparametric spectral density function estimator. Our numerical results show that the proposed spectral density estimator performs comparable with the parametric counterpart when the underlying process is assumed to be band‐limited. Additionally, this method is applied to analyze US Housing Starts Data, where the hidden periodicities are detected, providing consistent conclusions with previous economic studies.

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

This paper is concerned with robust performance analysis of linear stochastic systems acting as an integral operator on a standard Wiener process at the input and producing a stationary Gaussian random process at the output. We propose a performance criterion which uses the trace of an analytic function of the spectral density of the output process. This class of “covariance-analytic” cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the “cost-shaping” analytic function, the covariance-analytic performance criterion depends on higher-order Hardy-Schatten norms of the system transfer function. We discuss the links of these norms with the asymptotic behaviour of cumulants of finite-horizon quadratic functionals of the system output and their variational properties pertaining to system robustness to statistically uncertain inputs which differ from the standard Wiener process. In the case of strictly proper finite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of mixed products of noncommuting annihilation and creation operators in quantum mechanics. This computational procedure, which is one of the main results of the paper, involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared H2-norm of an auxiliary cascaded system. These results are also compared with an alternative approach, which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations, and illustrated by a numerical example.

The higher-order analysis of response for linear structure under non-Gaussian stochastic excitations with known statistical moments and power spectral density

The frequency-domain analysis method is a fundamental component of the random vibration analysis, in which the corresponding moment spectrum of excitations are the prerequisite. Nevertheless, the determination of the higher-order moment spectra for non-Gaussian stochastic excitations continues to pose a significant challenge in the existing research works. This paper introduces an accurate and efficient computational approach for determining higher-order moment spectra of non-Gaussian stochastic excitations with known statistical moments and power spectral density (PSD). Firstly, following the idea of the simulation method of non-Gaussian stochastic processes, the transformation model between the stationary non-Gaussian stochastic process and underlying Gaussian stochastic process is determined based on the known statistical moments, the PSD of the underlying Gaussian stochastic process is determined using the transformation model. Secondly, the approximate higher-order moment spectra models of stationary non-Gaussian stochastic processes are presented by the PSD of the underlying Gaussian process. Subsequently, the higher-order moment spectra models of non-Gaussian excitations are utilized to compute the higher-order moment spectra of response for the linear structure via the auxiliary harmonic excitation generalized method. Finally, three numerical examples are examined to assess the efficiency and accuracy of the proposed approximate models for the higher-order moment spectra of non-Gaussian stochastic processes.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

Forecasting of Male Life Expectancy in Nigeria: Box-Jenkins Approach

Over the past few decades, Males’ life expectancy has been observed to be less than that of females. In the literature, little has been done to forecast the Life expectancy at birth for Males (LEM) in developing countries, especially in Nigeria. To understand the forecasting dynamics of LEM in Nigeria, this work therefore examines the series using the Box-Jenkins (1976) methodology. Pre-examination tests conducted on the series revealed that it is not only a difference stationary process of order one {I(1)} but it is also best fitted by ARIMA (1, 1, 1) model. Moreover, diagnostic checking carried out on the residuals of the fitted model confirmed that the residual is a white noise error term in that it fulfils the assumptions of homoscedasticity, stationarity, and autocorrelation. To obtain a good forecast, the LEM data spanning 1960 to 2014 (55 years) were used to train the model while the remaining LEM data spanning 2015 to 2020 (6 years) were used to test the data. For the test data, the forecasted LEM has been shown to give approximate values of the actual LEM. Findings showed that the LEM is expected to increase slightly by 0.64% within the forecasted periods spanning 2021 to 2030. Diagnostic checking, Forecasting, Homoscedasticity, Life expectancy, Male, Training sets, Testing sets

Conditional simulation of stationary non-Gaussian processes based on unified hermite polynomial model

The conditional simulation of non-Gaussian excitations utilizing records from the monitoring system is of great significance for hazard mitigation. To this end, this paper proposes a novel conditional non-Gaussian simulation method. In this method, the Unified Hermite Polynomial Model (UHPM) is used to describe the transformation relationship between recorded and unrecorded non-Gaussian processes and their underlying Gaussian counterparts. Meanwhile, an explicit transformation model between their correlation functions is also provided. Then, the covariance matrix of Fourier coefficients of the underlying Gaussian processes is constructed. Based on this covariance matrix, the conditional samples of Fourier coefficients are generated and substituted into the Spectral Representation Method (SRM) to perform the conditional simulation of the underlying Gaussian processes. Finally, the conditionally simulated samples of the underlying Gaussian processes are transformed into the non-Gaussian samples by the UHPM. To showcase the precision and efficacy of the proposed method, two numerical examples involving the conditional simulations of non-Gaussian ground motions and non-Gaussian wind pressure coefficients are provided.

Intermittency as a Consequence of a Stationarity Constraint on the Energy Flux.

In his seminal work on turbulence, Kolmogorov made use of the stationary hypothesis to determine the power density spectrum of the velocity field in turbulent flows. However, to our knowledge, the constraints that stationary processes impose on the fluctuations of the energy flux have never been used in the context of turbulence. Here, we recall that the power density spectra of the fluctuations of the injected power, the dissipated power, and the energy flux have to converge to a common value at vanishing frequency. Hence, we show that the intermittent Gledzer-Ohkitani-Yamada (GOY) shell model fulfills these constraints. We argue that they can be related to intermittency. Indeed, we find that the constraint on the fluctuations of the energy flux implies a relation between the scaling exponents that characterize intermittency, which is verified by the GOY shell model and in agreement with the She-Leveque formula. It also fixes the intermittency parameter of the log-normal model at a realistic value. The relevance of these results for real turbulence is drawn in the concluding remarks.

Models for the estimation of gust-buffeting response in symmetrical wind-sensitive structures

The peak factor is an analytical development for estimating the expected maximum response caused by aerodynamic forces on wind-sensitive structures. It expresses the likelihood of how above the mean is the maximum amplitude of a stationary process in terms of its standard deviation. Such a process represents the structural response of a wind-sensitive structure in conventional applications. The methods and solutions used to approximate it in the along-wind direction have become ordinary to structural engineers. Notwithstanding, the structural response in combined orthogonal directions is often omitted since a criterion for such an estimation is still absent. This communication seeks to contribute to fill the gap in practical criteria for estimating response maxima in symmetric structures. It is done by proposing models to estimate the parameters describing the probability distribution of the maxima from the Euclidean norm of two correlated Gaussian variables. Although the applications to structural engineering are emphasized, the proposed models apply to other problems where the maxima from the norm of two Gaussian variables is relevant.

TWO-CARDINAL DERIVED TOPOLOGIES, INDESCRIBABILITY AND RAMSEYNESS

Abstract We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

Analyzing the degree persistence of shocks to energy security of the G7 countries: Evidence using panel SPSM-quantile unit root test

Energy security is affected by extreme natural, human, domestic political, geopolitical, and fossil energy price shocks/events and green energy policies. The degree of persistence in energy security determines the magnitudes of social, economic, and environmental outcomes of the shocks/policies. In this paper, we examined the degree of persistence in energy security of G7 countries using a new proxy namely the energy security risk index, and a novel second-generation panel quantile unit root test over the period 1980–2018. In addition, we applied the sequential panel selection method (SPSM), to identify the stationary members within each quantile. Our results indicated the stochastic properties of the energy security risk indexes vary across the quantile and the countries. Among the G7 countries, the energy security risk index of the US displays unit root process within all quantiles. While the energy security risk indexes of other countries display stationary processes, especially within high quantiles. Our results have important policy implications regarding the effectiveness of green policies in improving the energy security of the G7 countries and the disturbance effects of anti-energy security shocks. According to our findings, the US has to constantly pursue the risks that threaten the country's energy system while other G7 countries likely do not have such severe concerns about shocks affecting energy security, and these shocks have a short-term effect on their energy security.

Inflation dynamics and persistence: The importance of the uncertainty channel

In this article, we employ a time-varying GARCH-type specification to model inflation and investigate the behaviour of its persistence. Specifically, by modelling the inflation series as AR(1)-APARCH(1,1)-in-mean-level process with breaks, we show that persistence is transmitted from the conditional variance to the conditional mean. Accordingly, we propose a new measure of time-varying persistence, which not only distinguishes between changes in the dynamics of inflation and its volatility but also allows for feedback between the two variables. Analysing the inflation series for a number of countries, we find evidence that inflation uncertainty plays an important role in shaping expectations, and a higher level of uncertainty increases inflation persistence. We also consider a number of unit root tests and present the results of a Monte Carlo experiment to investigate the size and power properties of these tests in the presence of breaks in the mean and the variance equation of an AR(1)-APARCH(1,1)-in-mean-level data generating process. The Monte Carlo experiment reveals that if the model is misspecified, then commonly used unit root tests will misclassify inflation as a nonstationary, rather than a stationary process.